Finite Group Databasebeta

← #6493635

Subgroups — #6493635

Order 512.

2182 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#33110C214yesmaximal, complemented
4#29927C214yesmaximal, complemented
5#27978C212yesmaximal, complemented
6#27913C212yesmaximal, complemented
7#27895C214yesmaximal, complemented
8#27583C212yesmaximal, complemented
9#27582C212yesmaximal, complemented
10#24006C212yesmaximal, complemented
11#24005C212yesmaximal
12#21986C212yesmaximal, complemented
13#21808C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic, complemented
15C22×#124C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117—21no—
19C23×C42C2211yescharacteristic
20C2×#484C2214yescomplemented
21C2×#481C2214yes—
22C2×#432C2212yes—
23#2850—21no—
24#2850—21no—
25#2850—21no—
26#2850—21no—
27#2850C2212yes—
28#2850C2212yes—
29#2850C2212yes—
30#2850C2212yes—
31C2×#391C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#388—21no—
34C2×#388—21no—
35C2×#386—21no—
36C2×#386—21no—
37#2285—21no—
38#2285—21no—
39#2285—21no—
40#2285—21no—
41#2285C2212yes—
42#2285C2212yes—
43#2285C2212yescomplemented
44#2285C2212yescomplemented
45C2×#384C2211yescharacteristic, complemented
46C2×#384C2211yescharacteristic, complemented
47C2×#382—21no—
48C2×#382C2211yescharacteristic
49C2×#380—21no—
50C2×#380C2212yes—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2212yescomplemented
53#2096C2214yescomplemented
54#2093C2214yescomplemented
55#2072C2214yescomplemented
56#2071C2214yescomplemented
57#2036C2218yescomplemented
58#2034C2218yes—
59#1983C2218yescomplemented
60#1959C2214yescomplemented
61#1959C2214yescomplemented
62#1949C2218yescomplemented
63#1927C2218yescomplemented
64#1925C2218yescomplemented
65#1874C2214yes—
66#1872C2214yescomplemented
67#1853C2214yes—
68#1852C2214yes—
69#1847C2214yescomplemented
70#1836C2214yescomplemented
71#1817C2212yes—
72#1797C2212yescomplemented
73#1797C2212yescomplemented
74#1787C2214yes—
75#1786C2214yes—
76#1779C2212yes—
77#1749C2212yescomplemented
78#1749C2212yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4D411yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#124—21no—
86C2×#124—21no—
87C2×#124—28no—
88C2×#124C2312yes—
89C2×#124C2312yes—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—44no—
93C2×C22⋊Q8C2312yes—
94C2×C22⋊Q8C2312yes—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—44no—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117—48no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C42—28no—
122C22×C42—22no—
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#484—48no—
136#481—48no—
137#432—28no—
138C2×#107C2318yes—
139C2×#106C2318yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2318yes—
142C2×#103C2318yes—
143C2×#102C2318yes—
144C2×#102C2318yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×#97C2314yes—
149#391—21no—
150#391—21no—
151#391C2312yes—
152#391C2312yes—
153#390—21no—
154#390—21no—
155#390C2312yes—
156#390C2312yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#386—22no—
166#386—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#384—21no—
174#384—21no—
175#384—21no—
176#384—21no—
177#384C2312yes—
178#384C2312yes—
179#384C2312yes—
180#384C2312yes—
181#382—21no—
182#382—21no—
183#382—22no—
184#382—22no—
185#382—22no—
186#382—22no—
187#382C2312yes—
188#382C2312yes—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380C2314yes—
196#380C2314yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2314yes—
205C2×#95—22no—
206C2×#95—22no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215C25—21no—
216C25C2×D411yescharacteristic
217C22×Q8—21no—
218C22×Q8—24no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8C2412yes—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—24no—
226C22×D4—42no—
227C22×D4—42no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4C2412yes—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C22×C8C2418yes—
265#124—28no—
266#124—28no—
267#124—44no—
268#124—44no—
269C22⋊Q8—28no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273#121—28no—
274#121—44no—
275#121—44no—
276#121—44no—
277#117—48no—
278#117—48no—
279#117—48no—
280#117—48no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2414yes—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—48no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42#4118yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—48no—
361#106—48no—
362C4⋊C8—28no—
363#103—48no—
364#103—48no—
365#102—48no—
366#102—48no—
367C22⋊C8—28no—
368C22⋊C8—28no—
369C22⋊C8—44no—
370#97—28no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—41no—
389C24—41no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24#13714yes—
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C24C22×D411yescharacteristic
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—88no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—88no—
421C2×D4—816no—
422C22×C4—232no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—416no—
478C22×C4—416no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4#12314yes—
495C22×C4#12314yes—
496C22×C4AGL(1,ℤ/8)14yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4114yes—
509C22×C4H(5,𝔽2)14yes—
510C22×C4C2511yescharacteristic
511C2×C8—28no—
512C2×C8—28no—
513C2×C8—28no—
514C4⋊C4—28no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—28no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—48no—
559C42—48no—
560C42#12314yes—
561C42#12314yes—
562C42C2×#4114yes—
563C42C2×#4114yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—28no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—28no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23C2×#12312yes—
600C23#53512yes—
601C23C2×AGL(1,ℤ/8)12yes—
602C23C2×#13712yes—
603C23C2×#13712yes—
604C23C2×#13712yes—
605C23#57512yes—
606C23C23×D411yescharacteristic
607Q8—48no—
608Q8—84no—
609D4—48no—
610D4—84no—
611D4—88no—
612D4—88no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—88no—
671C2×C4—88no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4C2×#12312yes—
676C2×C4C2×#12312yes—
677C2×C4C2×#12312yes—
678C2×C4C2×#12312yes—
679C2×C4C2×#12312yes—
680C2×C4#53512yes—
681C2×C4C2×AGL(1,ℤ/8)12yes—
682C2×C4C2×AGL(1,ℤ/8)12yes—
683C2×C4#57512yes—
684C2×C4C22×#4112yes—
685C2×C4C2×H(5,𝔽2)12yes—
686C2×C4#58512yes—
687C8—48no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—416no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—168no—
716C22#309214yes—
717C22#321514yes—
718C22C22×#12311yescharacteristic
719C22C2×#53511yescharacteristic
720C22#344911yescharacteristic
721C22C22×AGL(1,ℤ/8)11yescharacteristic
722C22C22×#13711yescharacteristic
723C22C2×#57511yescharacteristic
724C22C2×#57511yescharacteristic
725C4—22no—
726C4—22no—
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—41no—
735C4—41no—
736C4—84no—
737C4—82no—
738C4—82no—
739C4—84no—
740C4—82no—
741C4#349414yes—
742C2—24no—
743C2—24no—
744C2—42no—
745C2—162no—
746C2#3526211yescharacteristic
747C2C2×#309211yescharacteristic
748C2#6051111yescharacteristic
749C2C2×#321511yescharacteristic
750C2#6080311yescharacteristic
751C2#6094911yescharacteristic
752C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2285C2semidirect
3. #33110C2semidirect
4. #29927C2semidirect
5. #27978C2semidirect
6. #27913C2semidirect
7. #27895C2semidirect
8. #27583C2semidirect
9. #27582C2semidirect
10. #24006C2semidirect
12. #21986C2semidirect
13. #21808C2semidirect
14. #20715C2semidirect
20. C2×#484C22semidirect
43. #2285C22semidirect
44. #2285C22semidirect
45. C2×#384C22semidirect
46. C2×#384C22semidirect
52. C2×C4×C4⋊C4C22semidirect
53. #2096C22semidirect
54. #2093C22semidirect
55. #2072C22semidirect
56. #2071C22semidirect
57. #2036C22semidirect
59. #1983C22semidirect
60. #1959C22semidirect
61. #1959C22semidirect
62. #1949C22semidirect
63. #1927C22semidirect
64. #1925C22semidirect
66. #1872C22semidirect
69. #1847C22semidirect
70. #1836C22semidirect
72. #1797C22semidirect
73. #1797C22semidirect
77. #1749C22semidirect
78. #1749C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#28502
C2×#22852
#331102
#299272
#279782
#279132
#278952
#275832
#275822
#240062
#240052
#219862
#218082
#207152